Problem: Let $f$ be a 2D scalar field. Is the expression $\text{curl} (\text{grad}(f))$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Solution: The gradient, which takes a scalar field and gives a vector field of the same dimension, can be written in two ways: $\text{grad}(f) = \nabla f$ The 2D curl, which takes a vector field and gives a scalar field, can be written in two ways: $\text{curl}(F) = \nabla \times F$ Therefore, $\text{curl} (\text{grad}(f))$ is the curl of the gradient of a 2D scalar field. The gradient of a 2D scalar field is a 2D vector field. The curl of a 2D vector field is a scalar field. The expression $\text{curl} (\text{grad}(f))$ is a scalar field.